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Here's the reason it should work: alot of money from the TNC-government transactions is lost to the actual people overseeing the deals, ie public servants. This does the authoritarian no-good, because this is money he could use to bribe his patrons who keep him in power. Instead, it goes to people who could conceivably challenge government power. This is even plausibly a part of the connection between abundant natural resources and civil war. So answer: reducing corruption actually results in the government getting more money (corruption that isn't theirs, that is).
But what to DO with this new money?
Well as always, it will be distributed on the basis of what "buys" the government more power. Military officers, tribe members, ministers, unionists, ect. BUT, the government has a self-interested reason to "bribe" the common people (this is known as public spending).
reason one: fear of civil uprising, civil war and terrorist attacks
reason two: diversifying the economy off mineral trade will, in some sectors, require alot of labor input. So winning the people over will help that go smoother.
The reason this is the Machiavellian solution is because it harks back to one of his principles: is it better to be feared or loved? well if at all possible, always be both!